# Definition:Möbius Strip

## Definition

A Möbius strip is a surface with boundary obtained by twisting a side of a rectangle by $180$ degrees, then joining the twisted side and the opposite side of it together.

Thus in the above diagram, $AB$ is identified with $CD$.

### Formal Construction

Let $T$ be the square embedded in the Cartesian plane defined as:

$T = \closedint 0 1 \times \closedint 0 1$

Let $T'$ be the quotient space formed from $T$ using the identification mapping $p: T \to T'$ as follows:

$\forall \tuple {x, y} \in T: \map p {x, y} = \begin {cases} \tuple {1, 1 - y} & : x = 0 \\ \paren {x, y} & : x \ne 0 \end {cases}$

## Also known as

Some sources (particularly American) use the term Möbius band on the grounds that the word strip may be considered inappropriate language to use in front of children and adolescents.

## Also see

• Results about the Möbius strip can be found here.

## Source of Name

This entry was named for August Ferdinand Möbius.